We consider the classical Minimum Crossing Number problem: given an $n$-vertex graph $G$, compute a drawing of $G$ in the plane, while minimizing the number of crossings between the images of its edges. This is a fundamental and extensively studied problem, whose approximability status is widely open. In all currently known approximation algorithms, the approximation factor depends polynomially on $\Delta$ -- the maximum vertex degree in $G$. The best current approximation algorithm achieves an $O(n^{1/2-\varepsilon}\cdot \text{poly}(\Delta\cdot\log n))$-approximation, for a small fixed constant $\epsilon$, while the best negative result is APX-hardness, leaving a large gap in our understanding of this basic problem. In this paper we design a randomized $O\left(2^{O((\log n)^{7/8}\log\log n)}\cdot\text{poly}(\Delta)\right )$-approximation algorithm for Minimum Crossing Number. This is the first approximation algorithm for the problem that achieves a subpolynomial in $n$ approximation factor (albeit only in graphs whose maximum vertex degree is subpolynomial in $n$). In order to achieve this approximation factor, we design a new algorithm for a closely related problem called Crossing Number with Rotation System, in which, for every vertex $v\in V(G)$, the circular ordering, in which the images of the edges incident to $v$ must enter the image of $v$ in the drawing is fixed as part of the input. Combining this result with the recent reduction of [Chuzhoy, Mahabadi, Tan '20] immediately yields the improved approximation algorithm for Minimum Crossing Number. We introduce several new technical tools, that we hope will be helpful in obtaining better algorithms for the problem in the future.

翻译：我们考虑古典最低跨度数字问题:考虑到一个$n 的顶端图形$G$,在平面上计算一张$G$的图画,同时尽量减少其边缘图像之间的交叉次数。这是一个经过广泛研究的根本性问题,其接近性状态是广泛开放的。在所有目前已知的近似算法中,近似系数多寡取决于$\Delta$ -- -- 最高顶顶点为$G$。在本文中,目前最佳近似算法将达到一个$(n ⁇ 1/2-\vareplacelón ⁇ cdot\ text$}(Delta\cdolog n}(Delta\ct\log n)$ 。对于一个小固定的固定不变的 $(eeplimexlity), 最接近的离差因子值(xxxxxxxx) 将自动设计一个 lexleft($%%%O ((() = lix nv=8/8) log nqot) listal{poly{t}t$(Dral) listal dalendation) numalational dal dal) 问题,这个在最近点上, irstal irstal irstal irmaxxxxxxxxxxxxxx 问题必须必须必须必须必须将立即将立即在新的一个新的排序。